Singular Dynamic Mode Decompositions. (arXiv:2106.02639v1 [eess.SY])

This manuscript is aimed at addressing several long standing limitations of
dynamic mode decompositions in the application of Koopman analysis. Principle
among these limitations are the convergence of associated Dynamic Mode
Decomposition algorithms and the existence of Koopman modes. To address these
limitations, two major modifications are made, where Koopman operators are
removed from the analysis in light of Liouville operators (known as Koopman
generators in special cases), and these operators are shown to be compact for
certain pairs of Hilbert spaces selected separately as the domain and range of
the operator. While eigenfunctions are discarded in this analysis, a viable
reconstruction algorithm is still demonstrated, and the sacrifice of
eigenfunctions realizes the theoretical goals of DMD analysis that have yet to
be achieved in other contexts. The manuscript concludes with the description of
a Dynamic Mode Decomposition algorithm that converges when a dense collection
of occupation kernels, arising from the data, are leveraged in the analysis.



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